Question

Please help me to prove this..​

Answer 1

We are given the equation 1 / sin(10°) - √3 / cos(10°) = 4, and want to prove that 1 / sin(10°) - √3 / cos(10°) equals 4.

Let's start by combining the expressions '1 / sin(10°)' and '√3 / cos(10°)' in the expression '1 / sin(10°) - √3 / cos(10°).'

1 / sin(10°) - √3 / cos(10°)

= cos(10°) - √3sin(10°) / sin(10°)
`*` cos(10°)

Now let's multiply the numerator and denominator by a common value. In this case it's most suitable to multiply both by 2. We will receive the expression 2(cos(10°) - √3sin(10°)) / 2(sin(10°)
`*` cos(10°)). We can further simplify this expression knowing that 2(sin(10°)
`*` cos(10°)) = 2sin(20°).

2(cos(10°) - √3sin(10°)) / 2sin(20°)

We can now bring the two on the bottom to the numerator, becoming 1 / 2. Remember that the ' 1 / 2 ' will be distributed now. After the distribution we receive the expression 4( 1 / 2(cos(10°) - √3 / 2sin(10°)) / 2sin(20°). We can now use the trivial functions 1 / 2 = sin(30°), and √3 / 2 = cos(30°).

4(sin(30°)(cos(10°) - cos(30°)(sin(10°) ) / sin(20°))

= 4((sin(30 - 10°)) / sin(20°) = 4

Answer 2

Explanation:

1 / (sin 10°) − √3 / (cos 10°)

Multiply top and bottom of the first fraction by ½ cos 10°.

½ cos 10° / (½ cos 10° sin 10°) − √3 / (cos 10°)

Multiply top and bottom of the second fraction by ½ sin 10°.

½ cos 10° / (½ cos 10° sin 10°) − ½√3 sin 10° / (½ sin 10° cos 10°)

Combine fractions.

(½ cos 10° − ½√3 sin 10°) / (½ sin 10° cos 10°)

Change ½ to sin 30°, and change ½√3 to cos 30°.

(sin 30° cos 10° − cos 30° sin 10°) / (½ sin 10° cos 10°)

Use angle sum formula.

sin(30° − 10°) / (½ sin 10° cos 10°)

sin 20° / (½ sin 10° cos 10°)

Multiply top and bottom by 4.

4 sin 20° / (2 sin 10° cos 10°)

Use double angle formula.

4 sin 20° / sin 20°

4